On the transfer of radiant energy

III;. J. Heat

Mass Tramfer.

Vol. 5, pp. 559-560. Pergamon Press 1962. Printed in Great Britain

LETTERS

TO THE EDITOR

ON THE TRANSFER

OF RADIANT

One cannot agree with Hottel that in the expression for D, the coefficient f should be used instead of $. As is known, the coefficient 4 relates to radiation which is isotropic over the whole space. The radiation considered in my paper (Fig. 1) is different and the coefficient f is not, therefore, appropriate. Finally, in my paper I did not make use of the concept of the effective thermal conductivity. Therefore the solutions obtained on the basis of it should not be identified with my solutions. P. K. KONAI
H. C. HOTTEL in his note “Radiation as a diffusion process” [l] compares an exact solution to the problem of radiant energy transfer in a plane layer of grey medium with the solutions given in my paper “On the regularities of composite heat transfer”. Comparison is made on the assumption that 1> I,, but it is then postulated that K is small. i.e. that the relation K = l/Is is invalid. Under these conditions, from formula (23) we obtain formula (26) with which an exact solution should be compared. If such a comparison is made, one may be convinced of a negligible divergence between both solutions. Hottel states that the equations I proposed do not permit of a distinction between T, and T: this is not so. The solution of equation (55) gives the relation T = p(x) whilst from equation (54) a relation between T, and T is obtained. Solutions for equations (54) and (55) were not given in my paper.

REPLY

TO

COMMENTS

M&JNICATION

BY ROGERS

“APPLICATION

OF THE AVERAGE

VELOCITY

AND

REFERENCE 1. H. C. HOTTEL, ht. J. Heat Mass Tratnfe~, 5. 82-83 (1962).

AND

OF THE

ENERGY

MAYHEW

DEFECT

ON

LAW

TEMPERATURE

TO

THE THE

SHORTER

COM-

DETERMINATION

IN TURBULENT

PIPE

FLOW”

BY W. SQUIRE* 1 MUST agree that the importance of the distinction between the average and bulk average temperatures was underestimated in my note [ 11. However, since the average is always less than the bulk average, a basic discrepancy remains, primarily in the liquid metal range. As Rogers and Mayhew point out in their letter, according to the conventional theories of turbulent heat transfer, the temperature distribution approaches the laminar form at sufficiently low Prandtl number, while the velocity distribution retains its turbulent form. The reason for this is that in the conventional analysis the ratio of eddy viscosity is found as a function of position by differentiation of the velocity profile. Then the eddv diffusivity is related to the eddy -viscosity. As a result, the eddy diffusivity is a multiple of the molecular viscosity, and becomes negligible compared to the molecular conductivity at sufficiently small Prandtl number. On the other hand, the assumption of a temperature defect profile, such as I made. imnlies that the temnerature profile will retain a turbulent character at low’Prandt1 number. In Fig. 1 I have plotted the temperature distribution * Received 15 February 1962. 559

0.8

0.6

< L

VI 0

I

I

0.2

I

I

04

II, 06

/, 0.8

I.0

Y/lo

FIG. 1. Theoretical

temperature Re = 108.

distribution

at